The sacrament of the Eucharist (Holy Communion) in the Roman Catholic Church and some other churches. Bread and wine are consecrated by a priest, and the elements (usually bread alone) distributed among the faithful. According to the doctrine of the Council of Trent (counteracting the teaching of the 16th-c Reformers) the bread and wine become the body and blood of Christ (transubstantiation), and the sacrament is to be understood as a divine, propitiatory sacrifice. Masses perform different functions in the life of the Church, eg a Requiem Mass for the dead, a Nuptial Mass for a marriage.
Unsolved problems in physics: What causes anything to have mass?Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. Mass is a central concept of classical mechanics and related subjects, and there are several forms of mass within the framework of relativistic kinematics (see mass in special relativity and mass in General Relativity). In the theory of relativity, the quantity invariant mass, which in concept is close to the classical idea of mass, does not vary between single observers in different reference frames. An object with small inertial mass changes its motion more readily, and an object with large inertial mass does so less readily. Within the same gravitational field, an object with a smaller passive gravitational mass experiences a smaller force than an object with a larger passive gravitational mass. In informal usage, the word "weight" is often used synonymously with "mass", because the strength of the gravitational field is roughly constant everywhere on the surface of the Earth. In physics, the two terms are distinct: an object will have a larger weight if it is placed in a stronger gravitational field, but its passive gravitational mass remains unchanged.) Active gravitational mass is a measure of the strength of the gravitational field due to a particular object. For example, the gravitational field that one experiences on the Moon is weaker than that of the Earth because the Moon has less active gravitational mass.
Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. The weak form of the equivalence principle states that this correspondence between inertial and gravitational masses is not accidental, and that no experiment will ever detect a difference between them.
Introduction
One of the consequences of the equivalence of inertial mass and passive gravitational mass is the fact, famously demonstrated by Galileo Galilei, that objects with different masses fall at the same rate, assuming factors like air resistance are negligible. The theory of general relativity, the most accurate theory of gravitation known to physicists to date, rests on the assumption that inertial and passive gravitational mass are completely equivalent. Classically, active and passive gravitational mass were equivalent as a consequence of Newton's third law, but a new axiom is required in the context of relativity's reformulation of gravity and mechanics. Thus, standard general relativity also assumes the equivalence of inertial mass and active gravitational mass;
If one were to treat inertial mass mi, passive gravitational mass mp, and active gravitational mass ma distinctly, Newton's law of universal gravitation would give as force on the second mass due to the first mass.
'Units of mass
In the SI system of units, mass is measured in kilograms (kg). Many other units of mass are also employed, such as: grams (g), tonnes, pounds, ounces, long and short tons, quintals, slugs, atomic mass units, Planck masses, solar masses, and eV/c2.
The eV/c (see below), it is possible to use any unit of energy as a unit of mass instead.
Because the gravitational acceleration (g) is approximately constant on the surface of the Earth, and also because mass-balances do not depend on the local value of g, a unit like the pound is often used to measure either mass or force (e.g. When the pound is used as a measure of mass (where g does not enter in), it is officially in the English system defined in terms of the kg, as 1 lb = 0.453 592 37 kg (see force.) In this case the English system unit of force is the poundal. By contrast, when the pound is used as the unit of force, the English unit of mass is the slug (mass).
For more information on the different units of mass, see Orders of magnitude (mass).
Inertial mass
Inertial mass is the mass of an object measured by its resistance to acceleration.
To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics.
According to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion
where f is the force acting on the body and v is its velocity.
Now, suppose that the mass of the body in question is a constant. This assumption, known as the conservation of mass, rests on the ideas that (i) mass is a measure of the amount of matter contained in a body, and (ii) matter can never be created or destroyed, only split up or recombined. Another point to note is that, even in classical mechanics, it is sometimes useful to treat the mass of an object as changing with time.
When the mass of a body is constant, Newton's second law becomes
where a denotes the acceleration of the body.
This equation illustrates how mass relates to the inertia of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. To be precise, suppose we have two objects A and B, with constant inertial masses mA and mB.
This is, in principle, how we would measure the inertial mass of an object. We choose a "reference" object and define its mass mB as (say) 1 kilogram. Then we can measure the mass of every other object in the universe by colliding it with the reference object and measuring the accelerations.
Gravitational mass
Gravitational mass is the mass of an object measured using the effect of a gravitational field on the object.
The concept of gravitational mass rests on Newton's law of gravitation. The law of gravitation states that if A and B have gravitational masses MA and MB respectively, then each object exerts a gravitational force on the other, of magnitude
where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the acceleration of a reference mass at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is
This is the basis by which masses are determined by weighing. In simple bathroom scales, for example, the force f is proportional to the displacement of the spring beneath the weighing pan (see Hooke's law), and the scales are calibrated to take g into account, allowing the mass M to be read off.
Equivalence of inertial and gravitational masses
The equivalence of inertial and gravitational masses is sometimes referred to as the Galilean equivalence principle or weak equivalence principle. Suppose we have an object with inertial and gravitational masses m and M respectively. If the only force acting on the object comes from a gravitational field g, combining Newton's second law and the gravitational law yields the acceleration
This says that the ratio of gravitational to inertial mass of any object is equal to some constant K if and only if all objects fall at the same rate in a given gravitational field.
Relativistic relation among mass, energy and momentum
See also: Mass in special relativitySpecial relativity is a necessary extension of classical physics.
In relativistic mechanics, the invariant mass (m) of a free particle is related to its energy (E) and momentum (p) by the equation
where c is the speed of light.
The first thing to notice about this equation is that it can cope with massless objects (m = 0), for which it reduces to
In classical mechanics, massless objects are an ill-defined concept, since applying any force to one would produce, via Newton's second law, an infinite acceleration.
Consider objects with non-zero mass. For these, the quantity m has a simple physical meaning: it is the inertial mass of the object as measured in its rest frame, the frame of reference in which its velocity is zero. The mass-energy-momentum relation thus reduces to
which states that the energy of an object as measured in its rest frame—its "rest energy"—is equal to its mass times the square of the speed of light.
Some books follow this up by stating that "mass and energy are equivalent", but this is somewhat misleading. The mass of an object, as we have defined it, is a quantity intrinsic to the object, and independent of our current frame of reference. it is true only in the rest frame of the object if the mass put into the equation is the rest mass or invariant mass.
Some authors define a quantity known as the relativistic mass, which is basically the quantity E/c to be true in all cases, and makes the "equivalence" of "mass" and energy true by definition, though neither quantity is frame-independent.
Having defined the mass of an object, let us look at how it behaves when not at rest. We can arrange the mass-energy-momentum relation in the following way:
When the momentum p is much smaller than mc, we can Taylor expand the square root, with the result
The leading term, which is the largest, is of course the rest energy.
For a macroscopic object, the rest energy mc2 includes the thermal energy, which depends on the temperature of the object, and is related to the random motion of the atoms or molecules of which the object is composed. In this case, the mass of system increases by a tiny amount if only the rest masses of the objects is (separately) considered, and not the fact that their kinetic energy contributes to the mass of the system (if all energies are considered, the mass of the system does not change when kinetic energy is converted to thermal energy). Similarly, metabolism, fire and other exothermic chemical processes convert mass to energy, however the mass change only appears after heat has been removed from the system, and even then is usually negligible. The reason is that mass, as we have defined it, is not conserved during such processes, if it is taken by summing rest energies or masses of system components (since this does not count the energy and mass associated with kinetic energy). However, during such transformations, mass continues to be conserved as a property of systems as invariant mass. However, the system of two photons continues to have the same mass as the system of electron-plus-positron, so system mass is conserved for any given observer. Other examples include nuclear fusion and nuclear fission, where system mass is conserved (the "rest mass" or invariant mass of the system as a whole), but the sum of rest masses of particles is not conserved. Energy, unlike the sum of rest masses, is always conserved in special relativity, so, roughly speaking, what is happening in these reactions is that the rest energy of the reactants is being transformed into the kinetic energy of the reaction products (though each kind of energy continues to contribute mass to the system).
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